Mercurial > matrix-functions
comparison toolbox/kms.m @ 2:c124219d7bfa draft
Re-add the 1995 toolbox after noticing the statement in the ~higham/mctoolbox/ webpage.
author | Antonio Pino Robles <data.script93@gmail.com> |
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date | Thu, 07 May 2015 18:36:24 +0200 |
parents | 8f23314345f4 |
children |
comparison
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1:e471a92d17be | 2:c124219d7bfa |
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1 function A = kms(n, rho) | |
2 %KMS Kac-Murdock-Szego Toeplitz matrix. | |
3 % A = KMS(N, RHO) is the N-by-N Kac-Murdock-Szego Toeplitz matrix with | |
4 % A(i,j) = RHO^(ABS((i-j))) (for real RHO). | |
5 % If RHO is complex, then the same formula holds except that elements | |
6 % below the diagonal are conjugated. | |
7 % RHO defaults to 0.5. | |
8 % Properties: | |
9 % A has an LDL' factorization with | |
10 % L = INV(TRIW(N,-RHO,1)'), | |
11 % D(i,i) = (1-ABS(RHO)^2)*EYE(N) except D(1,1) = 1. | |
12 % A is positive definite if and only if 0 < ABS(RHO) < 1. | |
13 % INV(A) is tridiagonal. | |
14 | |
15 % Reference: | |
16 % W.F. Trench, Numerical solution of the eigenvalue problem | |
17 % for Hermitian Toeplitz matrices, SIAM J. Matrix Analysis and Appl., | |
18 % 10 (1989), pp. 135-146 (and see the references therein). | |
19 | |
20 if nargin < 2, rho = 0.5; end | |
21 | |
22 A = (1:n)'*ones(1,n); | |
23 A = abs(A - A'); | |
24 A = rho .^ A; | |
25 if imag(rho) | |
26 A = conj(tril(A,-1)) + triu(A); | |
27 end |